Three dimensional blow-up solutions of the Navier Stokes equations

In this paper we extend the plane blow-up results of Grundy & McLaughlin (1997) to the three-dimensional Navier-Stokes equations. Using a solution structure originally due to Lin we first provide numerical evidence for the existence of blow-up solutions on -infinity < x, z < infinity, 0 less than or equal to y less than or equal to 1 with boundary conditions on y = 0 and y = 1 involving derivatives of the velocity components. The formulation enables us to consider plane and radial flow as special cases. Various features of the computations are isolated and are used to construct a formal asymptotic solution close to blow-up. We show that the numerical and asymptotic analyses provide a mutually consistent global picture which supports the conclusion that, for the family of problems we consider here, blow-up in fact can take place in three dimensions but at an inverse linear rate rather than the faster inverse square of the plane case.